This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of Martin-Loef type theory.Comment: 11 page

No abstract

The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer, which can be regarded as an algebraic formulation of type theory. We determine conditions for such models to satisfy the inference rules for dependent sums Σ, dependent products Π, and intensional identity types Id, as used in homotopy type theory. It is then shown that a category admits such a model if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: they should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class. It follows that many familiar settings for homotopy theory also admit natural models of the basic system of homotopy type theory.Homotopy type theory is an interpretation of constructive Martin-Löf type theory [20] into abstract homotopy theory. It allows type theory to be used as a formal calculus for reasoning about homotopy theory, as well as more general mathematics such as can be formulated in category theory or set theory under this new interpretation. Because constructive type theory has been implemented in computational proof assistants like Coq, homotopy type theory also facilitates the use of such computational tools in homotopy theory, category theory, set theory, and other fields of mathematics. This is just one aspect of the Univalent Foundations Program, which has recently been the object of quite intense investigation [24]. * Penultimate version; published as [2] 1 One thing missing from homotopy type theory, however, has been a notion of model that is both faithful to the precise formalism of type theory and yet general and flexible enough to be a practical tool for semantic investigations. Past attempts have involved either highly structured categories corresponding closely to the syntax of type theory, such as the categories with families of Dybjer [7], which are, however, somewhat impractical to work with semantically; or they use the more more natural and flexible setting of homotopical algebra, as in [4,6], but they must then be equipped (if possible) with structures satisfying unnatural coherence conditions, in order to model the type theory precisely.Here we present a new approach which attemps to combine the advantages of these two strategies. It is based on the observation that a category with families is the same thing as a representable natural transformation in the sense of Grothendieck. Ideas from Voevodsky [18] and are also used in an essential way. In the first section, the basic concept of a natural model is defined and shown to be adequate. The second section determines conditions for when the basic type constructors Σ, Π, Id are also modelled. This draws heavily on the methodology of [18]. Finally, the third section investigates the question of when a category admits such a model, concludin...

No abstract

The overall framework of our investigation is provided by topos theory. Direct proofs of the main results are given, but the specialist will recognize toposophical ideas in the background. Indeed, the duality between syntax and semantics is really a manifestation of that between algebra and geometry in the two directions of the geometric morphisms that lurk behind our formal theory. Along the way, we give an elementary proof of Butz and Moerdijk's result in logical terms.